RELATION BETWEEN CROSSED SQUARE AND CROSSED CORNER

Year 2001, Issue: 002, 89 - 96, 15.06.2001

Murat Alp Ahmet Bekir Erdal Ulualan

Abstract

The term crossed corner was introduced by Alp in (Alp , 1999) and its examples

were also given in (Alp . 1999). In this paper we who defined the crossed corner morphism

and also gave an important proposition which is established the equivalence between

crossed corner and crossed square.

Keywords

Crossed module, Cat'-group. Crossed Square, Crossed Comer, Gap

RELATION BETWEEN CROSSED SQUARE AND CROSSED CORNER

Abstract

The term crossed corner was introduced by Alp in (Alp , 1999) and its examples were also given in (Alp . 1999). In this paper we who defined the crossed corner morphism and also gave an important proposition which is established the equivalence between crossed corner and crossed square. The term of crossed module was introduced by J.H.C. Whitehead in (Whitehead . 1949). A computer programming package XMOD (Alp and Wensley. 2000) has been developed by C D. Wensley and M. Alp, written using the GAP (Schonert . 1993) group theory programming language to calculate crossed modules, their morphism and derivations; Cat1-groups, their morphism and sections. The study of bi-relative Steinberg groups has led to the definition of a type of 2-dimensional crossed module which is called crossed square in (Guin Walery and Loday , 1981). The term crossed corner which is a pair of crossed modules was defined and its some examples were given by Alp in (Alp , 1999) and (Alp . 2000) respectively. Section 2 contains some basic definitions such as crossed modules, crossed square, crossed comer and their standard examples. Section 3 includes a main theorem which gives the equivalent relation between crossed corner and crossed square.

Keywords

Crossed module, Cat'-group, Crossed Square, Crossed Comer, Gap

References

• ALP. M.. Characterization of Crossed corner , Algebras, Groups and Geometries , Vol 16. 173-182. (1999).
• ALP. M.. Some applications of Crossed corner. Algebras, Groups and Geometries, To appear
• ALP. M and WENSLEY, C. D. . XMOD Crossed modules and cat 1-groups in GAP. Manual for share package o f GAP, Chapter 73. 1357-1422, (1997)
• ALP. M and WENSLEY. C. D. , Enumeration of Cat'-groups of low order International Journal o f Algebra and Computation, Vol. 10, No. 4.407-424 . (2000).
• BROWN , R. and LODAY, J. L. , Van kampen theorems for diagrams of spaces I . Univ. Wales, Bangor, Preprint 1-69, (1984).
• ELLIS . G. J. , Crossed modules and their higher dimensional analogues. Univ. Wales , Bangor. Ph D Thesis , 1- 128 , (1984).
• GUiN WALERY. D. and LODAY , J. L. , Obstruction a l'e.xcision en Kllieoryalgebrique in Evanston conf. on Algebraic K-theory , Springer lectures Notes in Math.. 854 . 179-216 . (1981).
• LODAY . J. L. . Spaces with finitely many non-trivial homotopv groups . J. App. Algebra . 24 (1982) 179-202
• SCHÖNERT , M. ET AL , GAP: Groups, Algorithms, and Programming , Lehrstuhl D für Mathematik , Rheinisch Westfalische Technische Hoch-schule , Aachen , Gennany , third edition , 1993.
• WHITEHEAD , J. H. C. , On adding relations to homotopy groups, Ann. Math.. 47 . 806-810 , (1946).
• WHÎTEHEAD . J H. C. . Combinatorial homotopy II . Bull. A. M.S. , 55 (1949) 453-496.